Christian Espíndola: Infinitary generalizations of Deligne's completeness theorem

Abstract: Given a regular cardinal k such that k^<k=k (or any regular k if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points that we call the k-separable toposes. These are equivalent to sheaf toposes over a site with k-small limits that has at most k many objects and morphisms, the (basis for the) topology being generated by at most k many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough k-points, that is, points whose inverse image preserve all k-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when k is countable, when property T is trivially satisfied.

This result is essentially a completeness theorem for a certain infinitary logic that we call k-geometric, where conjunctions of less than k formulas and existential quantification on less than k many variables is allowed. We prove that k-geometric theories have a k-classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to k-geometric morphisms (geometric morphisms whose inverse image preserve all k-small limits) into that topos. Moreover, we prove that k-separable toposes occur as the k-classifying toposes of k-geometric theories of at most k many axioms in canonical form, and that every such k-classifying topos is k-separable.

Finally, we consider the case when k is weakly compact and study the k-classifying topos of a k-coherent theory (with at most k many axioms), that is, a theory where only disjunction of less than k formulas are allowed, obtaining a version of Deligne's theorem for k-coherent toposes from which we can derive, among other things, Karp's completeness theorem for infinitary classical logic.